# Dictionary Definition

sequence

### Noun

1 serial arrangement in which things follow in
logical order or a recurrent pattern; "the sequence of names was
alphabetical"; "he invented a technique to determine the sequence
of base pairs in DNA"

2 a following of one thing after another in time;
"the doctor saw a sequence of patients" [syn: chronological
sequence, succession, successiveness, chronological
succession]

3 film consisting of a succession of related
shots that develop a given subject in a movie [syn: episode]

4 the action of following in order; "he played
the trumps in sequence" [syn: succession]

5 several repetitions of a melodic phrase in
different keys

### Verb

1 arrange in a sequence

2 determine the order of constituents in; "They
sequenced the human genome"

# User Contributed Dictionary

## English

### Pronunciation

- /ˈsikwəns/ or /ˈsikwɛns/

### Noun

- A set of things next to each other in a set order; a series
- A series of musical phrases where a theme or melody is repeated, with some change each time, such as in pitch or length (example: opening of Beethoven's Fifth Symphony).
- A musical composition used in some Catholic Masses between the readings. The most famous sequence is the Dies Irae (Day of Wrath) formerly used in funeral services.
- An ordered list of objects.

#### Translations

set of things in a set order

series of musical phrases where a theme or
melody is repeated

poetic, music composition used in some Catholic
Masses between the readings

in mathematics, an ordered list of objects

- Czech: posloupnost
- Danish: følge
- Finnish: sarja

### Verb

- to arrange in an order
- to determine the order of things, especially of amino acids in a protein, or of bases in a nucleic acid

### Derived terms

# Extensive Definition

In mathematics, a sequence is
an ordered list of objects (or events). Like a set, it contains members
(also called elements or terms), and the number of terms (possibly
infinite) is called the length of the sequence. Unlike a set, order
matters, and the exact same elements can appear multiple times at
different positions in the sequence.

For example, (C, R, Y) is a sequence of letters
that differs from (Y, C, R), as the ordering matters. Sequences can
be finite, as in
this example, or infinite,
such as the sequence of all even
positive integers
(2, 4, 6,...).

## Examples and notation

There are various and quite different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the notations introduced below.A sequence may be denoted (a1, a2, ...). For
shortness, the notation (an) is also used.

A more formal definition of a finite sequence
with terms in a set S is a
function
from to S for some n ≥ 0. An infinite sequence in S is a function
from (the set of natural
numbers without 0) to S.

Sequences may also start from 0, so the first
term in the sequence is then a0.

A sequence of a fixed-length n is also called an
n-tuple.
Finite sequences include the empty sequence'' ( ) that has no
elements.

A function from all integers into a set is
sometimes called a bi-infinite sequence, since it may be thought of
as a sequence indexed by negative integers grafted onto a sequence
indexed by positive integers.

## Types and properties of sequences

A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements.If the terms of the sequence are a subset of an
ordered
set, then a monotonically increasing sequence is one for which
each term is greater than or equal to the term before it; if each
term is strictly greater
than the one preceding it, the sequence is called strictly
monotonically increasing. A monotonically decreasing sequence is
defined similarly. Any sequence fulfilling the monotonicity
property is called monotonic or monotone. This is a special case of
the more general notion of monotonic
function.

The terms non-decreasing and non-increasing are
used in order to avoid any possible confusion with strictly
increasing and strictly decreasing, respectively. If the terms of a
sequence are integers,
then the sequence is an integer
sequence. If the terms of a sequence are polynomials, then the
sequence is a polynomial
sequence.

If S is endowed with a topology, then it becomes
possible to consider convergence of an infinite sequence in S. Such
considerations involve the concept of the limit
of a sequence.

## Sequences in analysis

In analysis, when talking about sequences, one will generally consider sequences of the form- (x_1, x_2, x_3, \dots)\, or (x_0, x_1, x_2, \dots)\,

It may be convenient to have the sequence start
with an index different from 1 or 0. For example, the sequence
defined by xn = 1/log(n) would be defined only
for n ≥ 2. When talking about such infinite sequences, it is
usually sufficient (and does not change much for most
considerations) to assume that the members of the sequence are
defined at least for all indices large
enough, that is, greater than some given N.)

The most elementary type of sequences are
numerical ones, that is, sequences of real or complex
numbers. This type can be generalized to sequences of elements
of some vector
space. In analysis, the vector spaces considered are often
function
spaces. Even more generally, one can study sequences with
elements in some topological
space.

## Series

The sum of terms of a sequence is a series. More precisely, if (x1, x2, x3, ...) is a sequence, one may consider the sequence of partial sums (S1, S2, S3, ...), with- S_n=x_1+x_2+\dots + x_n=\sum\limits_^x_i.

- \sum\limits_^x_i.

## Infinite sequences in theoretical computer science

Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical computer science. They are often referred to simply as sequences (as opposed to finite strings). Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet ). The set C = ∞ of all infinite, binary sequences is sometimes called the Cantor space.An infinite binary sequence can represent a
formal
language (a set of strings) by setting the n th bit
of the sequence to 1 if and only if the n th string (in
shortlex
order) is in the language. Therefore, the study of complexity
classes, which are sets of languages, may be regarded as
studying sets of infinite sequences.

An infinite sequence drawn from the alphabet may
also represent a real number expressed in the base-b positional
number system. This equivalence is often used to bring the
techniques of real
analysis to bear on complexity classes.

## Sequences as vectors

Sequences over a field may also be viewed as vectors in a vector space. Specifically, the set of F-valued sequences (where F is a field) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers.In particular, the term sequence
space usually refers to a linear
subspace of the set of all possible infinite sequences with
elements in \mathbb.

## Doubly-infinite sequences

Normally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in the other -- the sequence has a first element, but no final element (a singly-infinite sequence). A doubly-infinite sequence is infinite in both directions -- it has neither a first nor a final element. Singly-infinite sequences are functions from the natural numbers (N) to some set, whereas doubly-infinite sequences are functions from the integers (Z) to some set.One can interpret singly infinite sequences as
element of the semigroup ring
of the natural
numbers R[\N], and doubly infinite sequences as elements of the
group
ring of the integers
R[\Z]. This perspective is used in the Cauchy
product of sequences.

## Ordinal-indexed sequence

An is a generalization of a sequence. If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. In this terminology an ω-indexed sequence is an ordinary sequence.## Sequences and automata

Automata or finite state machines can typically thought of as directed graphs, with edges labeled using some specific alphabet Σ. Most familiar types of automata transition from state to state by reading input letters from Σ, following edges with matching labels; the ordered input for such an automaton forms a sequence called a word (or input word). The sequence of states encountered by the automaton when processing a word is called a run. A nondeterministic automaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some input letter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence of single states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally used to mean the latter.## See also

- Net (topology) (a generalization of sequences)
- Sequence space
- Permutation
- Recurrence relation

### Types of sequences

### Related concepts

### Operations on sequences

## External links

sequence in Arabic: متتالية

sequence in Bulgarian: Редица

sequence in Catalan: Successió matemàtica

sequence in Danish: Talfølge

sequence in German: Folge (Mathematik)

sequence in Modern Greek (1453-):
Ακολουθία

sequence in Spanish: Sucesión matemática

sequence in Esperanto: Vico

sequence in Persian: توالی

sequence in French: Suite (mathématiques)

sequence in Galician: Sucesión
(matemáticas)

sequence in Korean: 수열

sequence in Croatian: Niz

sequence in Ido: Sequo

sequence in Indonesian: Barisan

sequence in Icelandic: Runa

sequence in Italian: Successione
(matematica)

sequence in Hebrew: סדרה

sequence in Hungarian: Sorozat
(matematika)

sequence in Dutch: Rij (wiskunde)

sequence in Japanese: 列 (数学)

sequence in Norwegian: Følge

sequence in Polish: Ciąg (matematyka)

sequence in Portuguese: Seqüência
matemática

sequence in Romanian: Şir (matematică)

sequence in Russian: Последовательность

sequence in Sicilian: Succissioni
(matimatica)

sequence in Simple English: Sequence

sequence in Slovenian: Zaporedje

sequence in Finnish: Lukujono

sequence in Swedish: Följd

sequence in Tamil: தொடர்வரிசை

sequence in Thai: ลำดับ

sequence in Vietnamese: Dãy (toán học)

sequence in Ukrainian: Послідовність
(математика)

sequence in Urdu: متوالیہ (ریاضی)

sequence in Chinese: 序列

# Synonyms, Antonyms and Related Words

Indian file, aftereffect, afterlife, aftermath, alternation, arrangement, array, articulation, bank, buzz, by-product, catena, catenation, chain, chain reaction, chaining, chasing, classification, concatenation, connectedness, connection, consecution, consecutiveness,
consequence,
consequent, consistency, continuity, continuum, corollary, course, cycle, degree, derivation, derivative, descent, development, disposal, disposition, distillate, distribution, dogging, drone, effect, endless belt, endless
round, event, eventuality, eventuation, file, filiation, following, fruit, future time, gamut, gradation, grouping, hangover, harvest, heeling, hierarchy, hounding, hum, issue, lateness, legacy, line, lineage, logical outcome,
monotone, next life,
nexus, offshoot, offspring, order, ordering, orderliness, organization, outcome, outgrowth, pendulum, periodicity, place, placement, plenum, postdate, postdating, posteriority, powder train,
precipitate,
procession, product, progression, provenience, pursual, pursuance, pursuit, queue, range, rank, recurrence, remainder, result, resultant, reticulation, rotation, round, routine, row, run, scale, sequel, sequela, sequent, serial order, series, set, shadowing, single file,
spectrum, string, subordination, subsequence, succession, supervenience, supervention, swath, system, tailing, thread, tier, trailing, train, upshot, windrow